# System Identification Based on Tensor Decompositions: A Trilinear Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background on Tensors

## 3. Trilinear Wiener Filter

## 4. LMS and NLMS Algorithms for Trilinear Forms

## 5. Simulation Results

#### 5.1. Iterative Wiener Filter

#### 5.2. LMS-TF and NLMS-TF

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Roy, R.; Sherman, J. A learning technique for Volterra series representation. IEEE Trans. Autom. Control
**1967**, 12, 761–764. [Google Scholar] [CrossRef] - Barker, H.A.; Obidegwu, S.N.; Pradisthayon, T. Performance of antisymmetric pseudorandom signals in the measurement of 2nd-order Volterra kernels by crosscorrelation. Proc. IEEE
**1972**, 119, 353–362. [Google Scholar] - Annabestani, M.; Naghavi, N.; Nejad, M.M. Nonautoregressive nonlinear identification of IPMC in large deformation situations using generalized Volterra-based approach. IEEE Trans. Instrum. Meas.
**2016**, 65, 2866–2872. [Google Scholar] [CrossRef] - Zhang, Z.; Ma, Y. Modeling of rate-dependent hysteresis using a GPO-based adaptive filter. Sensors
**2016**, 16, 205. [Google Scholar] [CrossRef] - Rugh, W.J. Nonlinear System Theory: The Volterra/Wiener Approach; Johns Hopkins University Press: Baltimore, MD, USA, 1981. [Google Scholar]
- Carassale, L.; Kareem, A. Modeling nonlinear systems by Volterra series. J. Eng. Mech.
**2010**, 136, 801–818. [Google Scholar] [CrossRef] - Benesty, J.; Paleologu, C.; Ciochină, S. On the identification of bilinear forms with the Wiener filter. IEEE Signal Process. Lett.
**2017**, 24, 653–657. [Google Scholar] [CrossRef] - Bai, E.-W.; Li, D. Convergence of the iterative Hammerstein system identification algorithm. IEEE Trans. Autom. Control
**2004**, 49, 1929–1940. [Google Scholar] [CrossRef] - Paleologu, C.; Benesty, J.; Ciochină, S. Adaptive filtering for the identification of bilinear forms. Digit. Signal Process.
**2018**, 75, 153–167. [Google Scholar] [CrossRef] - Dogariu, L.; Paleologu, C.; Ciochină, S.; Benesty, J.; Piantanida, P. Identification of bilinear forms with the Kalman filter. In Proceedings of the 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Calgary, AB, Canada, 15–20 April 2018; pp. 4134–4138. [Google Scholar]
- Dogariu, L.-M.; Ciochină, S.; Paleologu, C.; Benesty, J. A connection between the Kalman filter and an optimized LMS algorithm for bilinear forms. Algorithms
**2018**, 11, 211. [Google Scholar] [CrossRef] - Gesbert, D.; Duhamel, P. Robust blind joint data/channel estimation based on bilinear optimization. In Proceedings of the 8th Workshop on Statistical Signal and Array Processing, Corfu, Greece, 24–26 June 1996; pp. 168–171. [Google Scholar]
- Stenger, A.; Kellermann, W. Adaptation of a memoryless preprocessor for nonlinear acoustic echo cancelling. Signal Process.
**2000**, 80, 1747–1760. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, L.N.; Schwarz, S.; Rupp, M.; de Almeida, A.L.F.; Mota, J.C.M. A low-complexity equalizer for massive MIMO systems based on array separability. In Proceedings of the 2017 25th European Signal Processing Conference (EUSIPCO), Kos, Greece, 28 August–2 September 2017; pp. 2522–2526. [Google Scholar]
- Da Costa, M.N.; Favier, G.; Romano, J.M.T. Tensor modelling of MIMO communication systems with performance analysis and Kronecker receivers. Signal Process.
**2018**, 145, 304–316. [Google Scholar] [CrossRef] - Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice-Hall: Upper Saddle River, NJ, USA, 1999. [Google Scholar]
- Gay, S.L.; Benesty, J. (Eds.) Acoustic Signal Processing for Telecommunication; Kluwer Academic Publisher: Boston, MA, USA, 2000. [Google Scholar]
- Benesty, J.; Gaensler, T.; Morgan, D.R.; Sondhi, M.M.; Gay, S.L. Advances in Network and Acoustic Echo Cancellation; Springer: Berlin, Germany, 2001. [Google Scholar]
- Rupp, M.; Schwarz, S. A tensor LMS algorithm. In Proceedings of the 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, 19–24 April 2015; pp. 3347–3351. [Google Scholar]
- Rupp, M.; Schwarz, S. Gradient-based approaches to learn tensor products. In Proceedings of the 2015 23rd European Signal Processing Conference (EUSIPCO), Nice, France, 31 August–4 September 2015; pp. 2486–2490. [Google Scholar]
- Vervliet, N.; Debals, O.; Sorber, L.; Lathauwer, L.D. Breaking the curse of dimensionality using decompositions of incomplete tensors: Tensor-based scientific computing in big data analysis. IEEE Signal Process. Mag.
**2014**, 31, 71–79. [Google Scholar] [CrossRef] - Sidiropoulos, N.; Lathauwer, L.D.; Fu, X.; Huang, K.; Papalexakis, E.; Faloutsos, C. Tensor decomposition for signal processing and machine learning. IEEE Trans. Signal Process.
**2017**, 65, 3551–3582. [Google Scholar] [CrossRef] - Boussé, M.; Debals, O.; Lathauwer, L.D. A tensor-based method for large-scale blind source separation using segmentation. IEEE Trans. Signal Process.
**2017**, 65, 346–358. [Google Scholar] [CrossRef] - Lathauwer, L.D. Signal Processing Based on Multilinear Algebra. Ph.D. Thesis, Katholieke Universiteit Leuven, Leuven, Belgium, 1997. [Google Scholar]
- Kolda, T.G.; Bader, B.W. Tensor decompositions and applications. SIAM Rev.
**2009**, 51, 455–500. [Google Scholar] [CrossRef] - Comon, P. Tensors: A brief introduction. IEEE Signal Process. Mag.
**2014**, 31, 44–53. [Google Scholar] [CrossRef] - Cichocki, A.; Mandic, D.P.; Phan, A.H.; Caiafa, C.F.; Zhou, G.; Zhao, Q.; Lathauwer, L.D. Tensor decompositions for signal processing applications. IEEE Signal Process. Mag.
**2015**, 32, 145–163. [Google Scholar] [CrossRef] - Loan, C.F.V. The ubiquitous Kronecker product. J. Comput. Appl. Math.
**2000**, 123, 85–100. [Google Scholar] [CrossRef] [Green Version] - Paleologu, C.; Benesty, J.; Ciochină, S. Linear system identification based on a Kronecker product decomposition. IEEE/ACM Trans. Audio Speech Lang. Process.
**2018**, 26, 1793–1808. [Google Scholar] [CrossRef] - Elisei-Iliescu, C.; Paleologu, C.; Benesty, J.; Stanciu, C.; Anghel, C.; Ciochină, S. Recursive least-squares algorithms for the identification of low-rank systems. IEEE/ACM Trans. Audio Speech Lang. Process.
**2019**, 27, 903–918. [Google Scholar] [CrossRef] - Kiers, H.A.L. Towards a standardized notation and terminology in multiway analysis. J. Chemom.
**2000**, 14, 105–122. [Google Scholar] [CrossRef] - Kroonenberg, P. Applied Multiway Data Analysis; Wiley: Hoboken, NJ, USA, 2008. [Google Scholar]
- Ribeiro, L.N.; de Almeida, A.L.F.; Mota, J.C.M. Identification of separable systems using trilinear filtering. In Proceedings of the 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Cancun, Mexico, 13–16 December 2015; pp. 189–192. [Google Scholar]
- Morgan, D.R.; Kratzer, S.G. On a class of computationally efficient, rapidly converging, generalized NLMS algorithms. IEEE Signal Process. Lett.
**1996**, 3, 245–247. [Google Scholar] [CrossRef] - Haykin, S. Adaptive Filter Theory, 4th ed.; Prentice-Hall: Upper Saddle River, NJ, USA, 2002. [Google Scholar]
- Benesty, J.; Paleologu, C.; Ciochină, S. On regularization in adaptive filtering. IEEE Trans. Audio Speech Lang. Process.
**2011**, 19, 1734–1742. [Google Scholar] [CrossRef] - Digital Network Echo Cancellers; ITU-T Recommendations G.168; 2002. Available online: https://www.itu.int/rec/T-REC-G.168/en (accessed on 16 April 2019).
- Morgan, D.R.; Benesty, J.; Sondhi, M.M. On the evaluation of estimated impulse responses. IEEE Signal Process. Lett.
**1998**, 5, 174–176. [Google Scholar] [CrossRef]

**Figure 1.**Impulse responses used in simulations: (

**a**) ${\mathbf{h}}_{1}$ of length ${L}_{1}=64$ (the first impulse response from G168 Recommendation [37]), (

**b**) ${\mathbf{h}}_{2}$ of length ${L}_{2}=8$ (random impulse response with Gaussian distribution), (

**c**) ${\mathbf{h}}_{3}$ of length ${L}_{3}=4$ (its elements are evaluated as ${h}_{3{l}_{3}}={0.5}^{{l}_{3}-1},\phantom{\rule{4pt}{0ex}}{l}_{3}=1,\dots ,{L}_{3}$), and (

**d**) the global impulse response $\mathbf{h}={\mathbf{h}}_{3}\otimes {\mathbf{h}}_{2}\otimes {\mathbf{h}}_{1}$ of length $L={L}_{1}{L}_{2}{L}_{3}=2048$.

**Figure 2.**NM of the conventional Wiener filter as a function of the number of available data samples used to estimate the statistics (N), for the identification of the global impulse response from Figure 1d. The input signals are AR(1) processes, $L=2048$, and ${\sigma}_{v}^{2}=0.01$.

**Figure 3.**NM of the conventional and iterative Wiener filters, for different values of the number of available data samples used to estimate the statistics (N), for the identification of the global impulse response from Figure 1d. The input signals are AR(1) processes, $L=2048$, and ${\sigma}_{v}^{2}=0.01$.

**Figure 4.**NPM of the iterative Wiener filter, for different values of the number of available data samples used to estimate the statistics (N), for the identification of the individual impulse responses from Figure 1a–c: (

**a**) $\mathrm{NPM}\left({\mathbf{h}}_{1},{\widehat{\mathbf{h}}}_{1}^{\left(n\right)}\right)$, (

**b**) $\mathrm{NPM}\left({\mathbf{h}}_{2},{\widehat{\mathbf{h}}}_{2}^{\left(n\right)}\right)$, and (

**c**) $\mathrm{NPM}\left({\mathbf{h}}_{3},{\widehat{\mathbf{h}}}_{3}^{\left(n\right)}\right)$. The input signals are AR(1) processes, ${L}_{1}=64$, ${L}_{2}=8$, ${L}_{3}=4$, and ${\sigma}_{v}^{2}=0.01$.

**Figure 9.**NM of the NLMS-TF and regular NLMS algorithms. The impulse response ${\mathbf{h}}_{2}$ changes in the middle of the experiment.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dogariu, L.-M.; Ciochină, S.; Benesty, J.; Paleologu, C.
System Identification Based on Tensor Decompositions: A Trilinear Approach. *Symmetry* **2019**, *11*, 556.
https://doi.org/10.3390/sym11040556

**AMA Style**

Dogariu L-M, Ciochină S, Benesty J, Paleologu C.
System Identification Based on Tensor Decompositions: A Trilinear Approach. *Symmetry*. 2019; 11(4):556.
https://doi.org/10.3390/sym11040556

**Chicago/Turabian Style**

Dogariu, Laura-Maria, Silviu Ciochină, Jacob Benesty, and Constantin Paleologu.
2019. "System Identification Based on Tensor Decompositions: A Trilinear Approach" *Symmetry* 11, no. 4: 556.
https://doi.org/10.3390/sym11040556